An application of the Goulden-Jackson cluster theorem

移位麦克唐纳多项式与(q,t)变形的古尔德恩-杰克逊乘积

Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product

本文研究移位麦克唐纳多项式与(q,t)变形古尔德恩-杰克逊乘积，构造实现移位麦克唐纳本征值的无穷级数，推导纳扎罗夫-斯克莱宁算子相关级数并与已有算子比较。

A generalized Goulden-Jackson cluster method and lattice path enumeration

The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths---both with height bounded and unbounded---by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued f…

The top-degree part in the Matchings-Jack Conjecture

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{π, σ}^λ ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(α)}_π)$. The coefficients $ c_{π, σ}^λ $ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $ c_{π, σ}^λ $ are polynomials in the variable $β:= α-1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This \emph{Matching…

Annular noncrossing permutations and minimal transitive factorizations

We give two combinatorial proofs of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.

Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture

Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Dołęga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^λ_{μ^0,...,μ^k}(b)$ obtained by the expansion of thes…

基于Goulden-Jackson聚类定理的受限编码上界

Constrained coding upper bounds via Goulden-Jackson cluster theorem

本文利用Goulden-Jackson聚类定理提出计算受限编码精确基数的新方法，揭示了其与谱方法的内在联系，并给出了变长非重叠码的显式上界。

Double Hurwitz numbers via the infinite wedge

We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Goulden, Jackson and Vakil's Strong Piecewise Polynomiality conjecture. The method is a straightforward application of Okounkov's expression for double Hurwitz numbers in terms of operat…

Bounds on the frequency of 1 in the Kolakoski word

We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists, and establish the semi-rigorous bound |freq_1(K) - 1/2| <= 1/46.

Block patterns in generalized Euler Permutations

Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized clust…

Degree of W-operator and Noncrossing Partition

Goulden and Jackson first introduced the cut-and-join operator. The cut-and-join is widely used in studying the Hurwitz number and many other topological recursion problems. Mironov, Morosov and Natanzon give a more general construction and call it W-operator W([n]). As a special case, the cut-and-join operator is W([2]). In this paper, we study the structure of W([n]). We prove that W([n]) can be written as the sum of n! terms and each term corresponds uniquely to a permutation in Sn. We also define the degree of each term. We prove that there is a correspondence between the terms of W([n]) w…

Jackson-Richmond 4CT 常数恰好等于 10/27

The Jackson-Richmond 4CT Constant is EXACTLY 10/27

本文证明 Jackson 和 Richmond 在四色定理无计算机证明中使用的常数精确等于 10/27，并指出其评估有误，另给出若干类似但更复杂的结果。

Changes of variables in ELSV-type formulas

In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [7]. In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies. We also extend the results of [7] to include the Hodge integrals over the moduli spaces, in…

Constraining Strong c-Wilf Equivalence Using Cluster Poset Asymptotics

Let $π\in \mathfrak{S}_m$ and $σ\in \mathfrak{S}_n$ be permutations. An occurrence of $π$ in $σ$ as a consecutive pattern is a subsequence $σ_i σ_{i+1} \cdots σ_{i+m-1}$ of $σ$ with the same order relations as $π$. We say that patterns $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalent if for all $n$ and $k$, the number of permutations in $\mathfrak{S}_n$ with exactly $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In 2018, Dwyer and Elizalde conjectured (generalizing a conjecture of Elizalde from 2012) that if $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalen…

$q$-Analogues of $π$-Related Formulae from Jackson's $_8ϕ_7$-Series via Inversion Approach

By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfl…

与特定椭圆调和级数相关的度量及其Allouche-Hu-Morin极限定理

Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem

本文研究了特定椭圆调和级数的度量性质，并通过弱收敛性证明了Allouche-Hu-Morin定理，同时提供了误差估计和组合生成函数的分析。

交换幺半群的有界到一作用的超有限性

Hyperfiniteness of bounded-to-one actions of commutative monoids

该研究针对可数个两两交换的Borel函数生成等价关系的开放问题，在函数为有界到一的情形下给出肯定解答，推广了Gao-Jackson的相关定理。

New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions

We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2ϕ_1 \rightarrow {}_2ϕ_2$ transformation of Jackson to discover and prove many new weighted partition identities. These identities involve unrestricted partitions, overpartitions, and partitions with distinct even parts. Smallest part of the partitions plays an important role in our analysis. This work was motivated in part by the research of Krishna Alladi.

Multilateral inversion of A_r, C_r and D_r basic hypergeometric series

In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…

In Praise of an Elementary Identity of Euler

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. …